Archiv der Mathematik

, Volume 104, Issue 3, pp 283–288 | Cite as

Stability of dispersion managed solitons for vanishing average dispersion

  • Dirk Hundertmark
  • Peer Kunstmann
  • Roland Schnaubelt
Article

Abstract

We study dynamical properties of the dispersion management equation with vanishing average dispersion. Our main result establishes the stability of the set of ground states.

Keywords

Dispersion managed nonlinear Schrödinger equation Stability Ground states Compactness 

Mathematics Subject Classification

Primary 35Q55 Secondary 35B35 35Q60 

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References

  1. 1.
    Cazenave T., Lions P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85, 549–561 (1982)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Erdoğan M.B., Hundertmark D., Lee Y.-R.: Exponential decay of dispersion management solitons, Math. Res. Lett. 18, 11–24 (2011)MATHGoogle Scholar
  3. 3.
    Gabitov L., Turitsyn S.K.: Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation, Opt. Lett. 21, 327–329 (1996)Google Scholar
  4. 4.
    Gabitov L., Turitsyn S.K.: Breathing solitons in optical fiber links. JETP Lett. 63, 861 (1996)CrossRefGoogle Scholar
  5. 5.
    W.R. Green and D. Hundertmark, Exponential Decay for dispersion managed solitons for general dispersion profiles, Preprint, arXiv:1212.4004.
  6. 6.
    Hundertmark D., Lee Y.-R.: Decay estimates and smoothness for solutions of the dispersion managed non-linear Schrödinger equation, Comm. Math. Phys. 286, 851–873 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hundertmark D., Lee Y.-R.: On non-local variational problems with lack of compactness related to non-linear optics, J. Nonlinear Sci. 22, 1–38 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Kunze M.: On a variational problem with lack of compactness related to the Strichartz inequality, Calc. Var. Partial Differential Equations 19, 307–336 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Moloney and A. Newell, Nonlinear Optics, Westview Press, Boulder (CO), 2003.Google Scholar
  10. 10.
    Stanislalova M.: Regularity of ground state solutions of dispersion managed nonlinear Schrödinger equations, J. Differential Equations 210, 87–105 (2005)CrossRefMathSciNetGoogle Scholar
  11. 11.
    S.K. Turitsyn, E.G. Shapiro, S.B. Medvedev, M.P. Fedoruk, and V.K. Mezentsev, Physics and mathematics of dispersion-managed optical solitons, Comptes Rendus Physique 4 (2003), 145–161.Google Scholar
  12. 12.
    V. Zharnitsky, E. Grenier, C.K.R.T. Jones, and S.K. Turitsyn, Stabilizing effects of dispersion management, Phys. D 152–153 (2001), 794–817.Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Dirk Hundertmark
    • 1
  • Peer Kunstmann
    • 1
  • Roland Schnaubelt
    • 1
  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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